| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 1.46·11-s + 3.46·13-s − 15-s + 2·17-s + 1.46·19-s + 21-s + 6.92·23-s + 25-s − 27-s + 2·29-s + 1.46·31-s − 1.46·33-s − 35-s − 2·37-s − 3.46·39-s + 8.92·41-s − 4·43-s + 45-s − 2.92·47-s + 49-s − 2·51-s + 7.46·53-s + 1.46·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 0.333·9-s + 0.441·11-s + 0.960·13-s − 0.258·15-s + 0.485·17-s + 0.335·19-s + 0.218·21-s + 1.44·23-s + 0.200·25-s − 0.192·27-s + 0.371·29-s + 0.262·31-s − 0.254·33-s − 0.169·35-s − 0.328·37-s − 0.554·39-s + 1.39·41-s − 0.609·43-s + 0.149·45-s − 0.427·47-s + 0.142·49-s − 0.280·51-s + 1.02·53-s + 0.197·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.036243419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036243419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 1.07T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.87912209123383949180672900502, −7.17674344046504570419333261124, −6.40319569940687238379021658623, −6.01286381659231342896213378071, −5.20567834753200327901305140518, −4.49630783671953872135213074443, −3.53731082727911860070110246583, −2.85766804394961758494189301491, −1.58071059050447086514520473835, −0.816751008332718117402865207480,
0.816751008332718117402865207480, 1.58071059050447086514520473835, 2.85766804394961758494189301491, 3.53731082727911860070110246583, 4.49630783671953872135213074443, 5.20567834753200327901305140518, 6.01286381659231342896213378071, 6.40319569940687238379021658623, 7.17674344046504570419333261124, 7.87912209123383949180672900502
